Very simply, no, it was not possible under the law to be a conscientious objector. The conscription law that was enacted during the Meiji era had it's exemptions regarding other issues but none of that was applicable to a conscientious objector.

Later on those other exemptions were removed as well.

Now, regarding their individuals and their conscription. It's really up to the sect that how they would personally react. (There are at a minimum 18 major sects of Buddhism in war time Japan). And some of the sects like Nichiren aren't against their monks joining for the war effort. Some sects even donated money to the war effort. And in 1940, a new religion law was enacted that only government approved religion and sects are allowed. That put the final nail in the coffin for any pacifist sects.

And no matter what your personal objection is. You are going to the war whether you like it or not when the draft notice comes. (Along with a giant announcement by the Patriotic Women Society and a obligatory "Congratulations on your draft!" flag).

Just like in Germany, avoiding the draft is illegal and punishable offensive.